Let $y = x$ be the equation of a chord of the circle $\mathbf{C}_{1}$ (in the closed half-plane $x \

Question

Let $y = x$ be the equation of a chord of the circle $\mathbf{C}_{1}$ (in the closed half-plane $x \geq 0$ ) of diameter 10 passing through the origin. Let $C_{2}$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $C_{2}$, which passes through the point $(2,3)$ and is farthest from the center of $C_{2}$, is $x + ay + b = 0$, then $a - b$ is equal to :

Accepted Answer

Equation of circle $C_{2}$ is$$x^{2} + y^{2} - 5x - 5y = 0 $$its centre is $\left( \frac{5}{2},\frac{5}{2} ight)$$$m_{AB} = - 1 $$∴ Slope of required chord $= 1$∴ equation of required chord is $x - y + 1 = 0$$${	herefore a = - 1,	ext{ }b = 2 }{	herefore a - b = - 2}$$

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